// Copyright (c) 2021, gottingen group.
// All rights reserved.
// Created by liyinbin lijippy@163.com
//

// A btree implementation of the STL set and map interfaces. A btree is smaller
// and generally also faster than STL set/map (refer to the benchmarks below).
// The red-black tree implementation of STL set/map has an overhead of 3
// pointers (left, right and parent) plus the node color information for each
// stored value. So a set<int32_t> consumes 40 bytes for each value stored in
// 64-bit mode. This btree implementation stores multiple values on fixed
// size nodes (usually 256 bytes) and doesn't store child pointers for leaf
// nodes. The result is that a btree_set<int32_t> may use much less memory per
// stored value. For the random insertion benchmark in btree_bench.cc, a
// btree_set<int32_t> with node-size of 256 uses 5.1 bytes per stored value.
//
// The packing of multiple values on to each node of a btree has another effect
// besides better space utilization: better cache locality due to fewer cache
// lines being accessed. Better cache locality translates into faster
// operations.
//
// CAVEATS
//
// Insertions and deletions on a btree can cause splitting, merging or
// rebalancing of btree nodes. And even without these operations, insertions
// and deletions on a btree will move values around within a node. In both
// cases, the result is that insertions and deletions can invalidate iterators
// pointing to values other than the one being inserted/deleted. Therefore, this
// container does not provide pointer stability. This is notably different from
// STL set/map which takes care to not invalidate iterators on insert/erase
// except, of course, for iterators pointing to the value being erased.  A
// partial workaround when erasing is available: erase() returns an iterator
// pointing to the item just after the one that was erased (or end() if none
// exists).

#ifndef ABEL_CONTAINER_INTERNAL_BTREE_H_
#define ABEL_CONTAINER_INTERNAL_BTREE_H_

#include <algorithm>
#include <cassert>
#include <cstddef>
#include <cstdint>
#include <cstring>
#include <functional>
#include <iterator>
#include <limits>
#include <new>
#include <string>
#include <type_traits>
#include <utility>

#include "abel/base/profile.h"
#include "abel/container/internal/common.h"
#include "abel/container/internal/compressed_tuple.h"
#include "abel/container/internal/container_memory.h"
#include "abel/container/internal/layout.h"
#include "abel/memory/memory.h"
#include "abel/meta/type_traits.h"
#include <string_view>
#include "abel/container/compare.h"
#include "abel/utility/utility.h"

namespace abel {

namespace container_internal {

// A helper class that indicates if the Compare parameter is a key-compare-to
// comparator.
template<typename Compare, typename T>
using btree_is_key_compare_to =
std::is_convertible<abel::result_of_t<Compare(const T &, const T &)>,
        abel::weak_ordering>;

struct StringBtreeDefaultLess {
    using is_transparent = void;

    StringBtreeDefaultLess() = default;

    // Compatibility constructor.
    StringBtreeDefaultLess(std::less<std::string>) {}  // NOLINT
    StringBtreeDefaultLess(std::less<std::string_view>) {}  // NOLINT

    abel::weak_ordering operator()(std::string_view lhs,
                                   std::string_view rhs) const {
        return compare_internal::compare_result_as_ordering(lhs.compare(rhs));
    }
};

struct StringBtreeDefaultGreater {
    using is_transparent = void;

    StringBtreeDefaultGreater() = default;

    StringBtreeDefaultGreater(std::greater<std::string>) {}  // NOLINT
    StringBtreeDefaultGreater(std::greater<std::string_view>) {}  // NOLINT

    abel::weak_ordering operator()(std::string_view lhs,
                                   std::string_view rhs) const {
        return compare_internal::compare_result_as_ordering(rhs.compare(lhs));
    }
};

// A helper class to convert a boolean comparison into a three-way "compare-to"
// comparison that returns a negative value to indicate less-than, zero to
// indicate equality and a positive value to indicate greater-than. This helper
// class is specialized for less<std::string>, greater<std::string>,
// less<string_view>, and greater<string_view>.
//
// key_compare_to_adapter is provided so that btree users
// automatically get the more efficient compare-to code when using common
// google string types with common comparison functors.
// These string-like specializations also turn on heterogeneous lookup by
// default.
template<typename Compare>
struct key_compare_to_adapter {
    using type = Compare;
};

template<>
struct key_compare_to_adapter<std::less<std::string>> {
    using type = StringBtreeDefaultLess;
};

template<>
struct key_compare_to_adapter<std::greater<std::string>> {
    using type = StringBtreeDefaultGreater;
};

template<>
struct key_compare_to_adapter<std::less<std::string_view>> {
    using type = StringBtreeDefaultLess;
};

template<>
struct key_compare_to_adapter<std::greater<std::string_view>> {
    using type = StringBtreeDefaultGreater;
};

template<typename Key, typename Compare, typename Alloc, int TargetNodeSize,
        bool Multi, typename SlotPolicy>
struct common_params {
    // If Compare is a common comparator for a std::string-like type, then we adapt it
    // to use heterogeneous lookup and to be a key-compare-to comparator.
    using key_compare = typename key_compare_to_adapter<Compare>::type;
    // A type which indicates if we have a key-compare-to functor or a plain old
    // key-compare functor.
    using is_key_compare_to = btree_is_key_compare_to<key_compare, Key>;

    using allocator_type = Alloc;
    using key_type = Key;
    using size_type = std::make_signed<size_t>::type;
    using difference_type = ptrdiff_t;

    // True if this is a multiset or multimap.
    using is_multi_container = std::integral_constant<bool, Multi>;

    using slot_policy = SlotPolicy;
    using slot_type = typename slot_policy::slot_type;
    using value_type = typename slot_policy::value_type;
    using init_type = typename slot_policy::mutable_value_type;
    using pointer = value_type *;
    using const_pointer = const value_type *;
    using reference = value_type &;
    using const_reference = const value_type &;

    enum {
        kTargetNodeSize = TargetNodeSize,

        // Upper bound for the available space for values. This is largest for leaf
        // nodes, which have overhead of at least a pointer + 4 bytes (for storing
        // 3 field_types and an enum).
        kNodeValueSpace =
        TargetNodeSize - /*minimum overhead=*/(sizeof(void *) + 4),
    };

    // This is an integral type large enough to hold as many
    // ValueSize-values as will fit a node of TargetNodeSize bytes.
    using node_count_type =
    abel::conditional_t<(kNodeValueSpace / sizeof(value_type) >
                         (std::numeric_limits<uint8_t>::max)()),
            uint16_t, uint8_t>;  // NOLINT

    // The following methods are necessary for passing this struct as PolicyTraits
    // for node_handle and/or are used within btree.
    static value_type &element(slot_type *slot) {
        return slot_policy::element(slot);
    }

    static const value_type &element(const slot_type *slot) {
        return slot_policy::element(slot);
    }

    template<class... Args>
    static void construct(Alloc *alloc, slot_type *slot, Args &&... args) {
        slot_policy::construct(alloc, slot, std::forward<Args>(args)...);
    }

    static void construct(Alloc *alloc, slot_type *slot, slot_type *other) {
        slot_policy::construct(alloc, slot, other);
    }

    static void destroy(Alloc *alloc, slot_type *slot) {
        slot_policy::destroy(alloc, slot);
    }

    static void transfer(Alloc *alloc, slot_type *new_slot, slot_type *old_slot) {
        construct(alloc, new_slot, old_slot);
        destroy(alloc, old_slot);
    }

    static void swap(Alloc *alloc, slot_type *a, slot_type *b) {
        slot_policy::swap(alloc, a, b);
    }

    static void move(Alloc *alloc, slot_type *src, slot_type *dest) {
        slot_policy::move(alloc, src, dest);
    }

    static void move(Alloc *alloc, slot_type *first, slot_type *last,
                     slot_type *result) {
        slot_policy::move(alloc, first, last, result);
    }
};

// A parameters structure for holding the type parameters for a btree_map.
// Compare and Alloc should be nothrow copy-constructible.
template<typename Key, typename Data, typename Compare, typename Alloc,
        int TargetNodeSize, bool Multi>
struct map_params : common_params<Key, Compare, Alloc, TargetNodeSize, Multi,
        map_slot_policy<Key, Data>> {
    using super_type = typename map_params::common_params;
    using mapped_type = Data;
    // This type allows us to move keys when it is safe to do so. It is safe
    // for maps in which value_type and mutable_value_type are layout compatible.
    using slot_policy = typename super_type::slot_policy;
    using slot_type = typename super_type::slot_type;
    using value_type = typename super_type::value_type;
    using init_type = typename super_type::init_type;

    using key_compare = typename super_type::key_compare;

    // Inherit from key_compare for empty base class optimization.
    struct value_compare : private key_compare {
        value_compare() = default;

        explicit value_compare(const key_compare &cmp) : key_compare(cmp) {}

        template<typename T, typename U>
        auto operator()(const T &left, const U &right) const
        -> decltype(std::declval<key_compare>()(left.first, right.first)) {
            return key_compare::operator()(left.first, right.first);
        }
    };

    using is_map_container = std::true_type;

    static const Key &key(const value_type &x) { return x.first; }

    static const Key &key(const init_type &x) { return x.first; }

    static const Key &key(const slot_type *x) { return slot_policy::key(x); }

    static mapped_type &value(value_type *value) { return value->second; }
};

// This type implements the necessary functions from the
// abel::container_internal::slot_type interface.
template<typename Key>
struct set_slot_policy {
    using slot_type = Key;
    using value_type = Key;
    using mutable_value_type = Key;

    static value_type &element(slot_type *slot) { return *slot; }

    static const value_type &element(const slot_type *slot) { return *slot; }

    template<typename Alloc, class... Args>
    static void construct(Alloc *alloc, slot_type *slot, Args &&... args) {
        abel::allocator_traits<Alloc>::construct(*alloc, slot,
                                                 std::forward<Args>(args)...);
    }

    template<typename Alloc>
    static void construct(Alloc *alloc, slot_type *slot, slot_type *other) {
        abel::allocator_traits<Alloc>::construct(*alloc, slot, std::move(*other));
    }

    template<typename Alloc>
    static void destroy(Alloc *alloc, slot_type *slot) {
        abel::allocator_traits<Alloc>::destroy(*alloc, slot);
    }

    template<typename Alloc>
    static void swap(Alloc * /*alloc*/, slot_type *a, slot_type *b) {
        using std::swap;
        swap(*a, *b);
    }

    template<typename Alloc>
    static void move(Alloc * /*alloc*/, slot_type *src, slot_type *dest) {
        *dest = std::move(*src);
    }

    template<typename Alloc>
    static void move(Alloc *alloc, slot_type *first, slot_type *last,
                     slot_type *result) {
        for (slot_type *src = first, *dest = result; src != last; ++src, ++dest)
            move(alloc, src, dest);
    }
};

// A parameters structure for holding the type parameters for a btree_set.
// Compare and Alloc should be nothrow copy-constructible.
template<typename Key, typename Compare, typename Alloc, int TargetNodeSize,
        bool Multi>
struct set_params : common_params<Key, Compare, Alloc, TargetNodeSize, Multi,
        set_slot_policy<Key>> {
    using value_type = Key;
    using slot_type = typename set_params::common_params::slot_type;
    using value_compare = typename set_params::common_params::key_compare;
    using is_map_container = std::false_type;

    static const Key &key(const value_type &x) { return x; }

    static const Key &key(const slot_type *x) { return *x; }
};

// An adapter class that converts a lower-bound compare into an upper-bound
// compare. Note: there is no need to make a version of this adapter specialized
// for key-compare-to functors because the upper-bound (the first value greater
// than the input) is never an exact match.
template<typename Compare>
struct upper_bound_adapter {
    explicit upper_bound_adapter(const Compare &c) : comp(c) {}

    template<typename K, typename LK>
    bool operator()(const K &a, const LK &b) const {
        // Returns true when a is not greater than b.
        return !compare_internal::compare_result_as_less_than(comp(b, a));
    }

  private:
    Compare comp;
};

enum class MatchKind : uint8_t {
    kEq, kNe
};

template<typename V, bool IsCompareTo>
struct SearchResult {
    V value;
    MatchKind match;

    static constexpr bool HasMatch() { return true; }

    bool IsEq() const { return match == MatchKind::kEq; }
};

// When we don't use CompareTo, `match` is not present.
// This ensures that callers can't use it accidentally when it provides no
// useful information.
template<typename V>
struct SearchResult<V, false> {
    V value;

    static constexpr bool HasMatch() { return false; }

    static constexpr bool IsEq() { return false; }
};

// A node in the btree holding. The same node type is used for both internal
// and leaf nodes in the btree, though the nodes are allocated in such a way
// that the children array is only valid in internal nodes.
template<typename Params>
class btree_node {
    using is_key_compare_to = typename Params::is_key_compare_to;
    using is_multi_container = typename Params::is_multi_container;
    using field_type = typename Params::node_count_type;
    using allocator_type = typename Params::allocator_type;
    using slot_type = typename Params::slot_type;

  public:
    using params_type = Params;
    using key_type = typename Params::key_type;
    using value_type = typename Params::value_type;
    using pointer = typename Params::pointer;
    using const_pointer = typename Params::const_pointer;
    using reference = typename Params::reference;
    using const_reference = typename Params::const_reference;
    using key_compare = typename Params::key_compare;
    using size_type = typename Params::size_type;
    using difference_type = typename Params::difference_type;

    // Btree decides whether to use linear node search as follows:
    //   - If the key is arithmetic and the comparator is std::less or
    //     std::greater, choose linear.
    //   - Otherwise, choose binary.
    // TODO(ezb): Might make sense to add condition(s) based on node-size.
    using use_linear_search = std::integral_constant<
            bool,
            std::is_arithmetic<key_type>::value &&
            (std::is_same<std::less<key_type>, key_compare>::value ||
             std::is_same<std::greater<key_type>, key_compare>::value)>;

    // This class is organized by gtl::Layout as if it had the following
    // structure:
    //   // A pointer to the node's parent.
    //   btree_node *parent;
    //
    //   // The position of the node in the node's parent.
    //   field_type position;
    //   // The index of the first populated value in `values`.
    //   // TODO(ezb): right now, `start` is always 0. Update insertion/merge
    //   // logic to allow for floating storage within nodes.
    //   field_type start;
    //   // The count of the number of populated values in the node.
    //   field_type count;
    //   // The maximum number of values the node can hold. This is an integer in
    //   // [1, kNodeValues] for root leaf nodes, kNodeValues for non-root leaf
    //   // nodes, and kInternalNodeMaxCount (as a sentinel value) for internal
    //   // nodes (even though there are still kNodeValues values in the node).
    //   // TODO(ezb): make max_count use only 4 bits and record log2(capacity)
    //   // to free extra bits for is_root, etc.
    //   field_type max_count;
    //
    //   // The array of values. The capacity is `max_count` for leaf nodes and
    //   // kNodeValues for internal nodes. Only the values in
    //   // [start, start + count) have been initialized and are valid.
    //   slot_type values[max_count];
    //
    //   // The array of child pointers. The keys in children[i] are all less
    //   // than key(i). The keys in children[i + 1] are all greater than key(i).
    //   // There are 0 children for leaf nodes and kNodeValues + 1 children for
    //   // internal nodes.
    //   btree_node *children[kNodeValues + 1];
    //
    // This class is only constructed by EmptyNodeType. Normally, pointers to the
    // layout above are allocated, cast to btree_node*, and de-allocated within
    // the btree implementation.
    ~btree_node() = default;

    btree_node(btree_node const &) = delete;

    btree_node &operator=(btree_node const &) = delete;

    // Public for EmptyNodeType.
    constexpr static size_type Alignment() {
        static_assert(LeafLayout(1).Alignment() == InternalLayout().Alignment(),
                      "Alignment of all nodes must be equal.");
        return InternalLayout().Alignment();
    }

  protected:
    btree_node() = default;

  private:
    using layout_type = abel::container_internal::Layout<btree_node *, field_type,
            slot_type, btree_node *>;

    constexpr static size_type SizeWithNValues(size_type n) {
        return layout_type(/*parent*/ 1,
                /*position, start, count, max_count*/ 4,
                /*values*/ n,
                /*children*/ 0)
                .AllocSize();
    }

    // A lower bound for the overhead of fields other than values in a leaf node.
    constexpr static size_type MinimumOverhead() {
        return SizeWithNValues(1) - sizeof(value_type);
    }

    // Compute how many values we can fit onto a leaf node taking into account
    // padding.
    constexpr static size_type NodeTargetValues(const int begin, const int end) {
        return begin == end ? begin
                            : SizeWithNValues((begin + end) / 2 + 1) >
                              params_type::kTargetNodeSize
                              ? NodeTargetValues(begin, (begin + end) / 2)
                              : NodeTargetValues((begin + end) / 2 + 1, end);
    }

    enum {
        kTargetNodeSize = params_type::kTargetNodeSize,
        kNodeTargetValues = NodeTargetValues(0, params_type::kTargetNodeSize),

        // We need a minimum of 3 values per internal node in order to perform
        // splitting (1 value for the two nodes involved in the split and 1 value
        // propagated to the parent as the delimiter for the split).
        kNodeValues = kNodeTargetValues >= 3 ? kNodeTargetValues : 3,

        // The node is internal (i.e. is not a leaf node) if and only if `max_count`
        // has this value.
        kInternalNodeMaxCount = 0,
    };

    // Leaves can have less than kNodeValues values.
    constexpr static layout_type LeafLayout(const int max_values = kNodeValues) {
        return layout_type(/*parent*/ 1,
                /*position, start, count, max_count*/ 4,
                /*values*/ max_values,
                /*children*/ 0);
    }

    constexpr static layout_type InternalLayout() {
        return layout_type(/*parent*/ 1,
                /*position, start, count, max_count*/ 4,
                /*values*/ kNodeValues,
                /*children*/ kNodeValues + 1);
    }

    constexpr static size_type LeafSize(const int max_values = kNodeValues) {
        return LeafLayout(max_values).AllocSize();
    }

    constexpr static size_type InternalSize() {
        return InternalLayout().AllocSize();
    }

    // N is the index of the type in the Layout definition.
    // element_type<N> is the Nth type in the Layout definition.
    template<size_type N>
    ABEL_FORCE_INLINE typename layout_type::template element_type<N> *GetField() {
        // We assert that we don't read from values that aren't there.
        assert(N < 3 || !leaf());
        return InternalLayout().template Pointer<N>(reinterpret_cast<char *>(this));
    }

    template<size_type N>
    ABEL_FORCE_INLINE const typename layout_type::template element_type<N> *GetField() const {
        assert(N < 3 || !leaf());
        return InternalLayout().template Pointer<N>(
                reinterpret_cast<const char *>(this));
    }

    void set_parent(btree_node *p) { *GetField<0>() = p; }

    field_type &mutable_count() { return GetField<1>()[2]; }

    slot_type *slot(int i) { return &GetField<2>()[i]; }

    const slot_type *slot(int i) const { return &GetField<2>()[i]; }

    void set_position(field_type v) { GetField<1>()[0] = v; }

    void set_start(field_type v) { GetField<1>()[1] = v; }

    void set_count(field_type v) { GetField<1>()[2] = v; }

    // This method is only called by the node init methods.
    void set_max_count(field_type v) { GetField<1>()[3] = v; }

  public:
    // Whether this is a leaf node or not. This value doesn't change after the
    // node is created.
    bool leaf() const { return GetField<1>()[3] != kInternalNodeMaxCount; }

    // Getter for the position of this node in its parent.
    field_type position() const { return GetField<1>()[0]; }

    // Getter for the offset of the first value in the `values` array.
    field_type start() const { return GetField<1>()[1]; }

    // Getters for the number of values stored in this node.
    field_type count() const { return GetField<1>()[2]; }

    field_type max_count() const {
        // Internal nodes have max_count==kInternalNodeMaxCount.
        // Leaf nodes have max_count in [1, kNodeValues].
        const field_type max_count = GetField<1>()[3];
        return max_count == field_type{kInternalNodeMaxCount}
               ? field_type{kNodeValues}
               : max_count;
    }

    // Getter for the parent of this node.
    btree_node *parent() const { return *GetField<0>(); }

    // Getter for whether the node is the root of the tree. The parent of the
    // root of the tree is the leftmost node in the tree which is guaranteed to
    // be a leaf.
    bool is_root() const { return parent()->leaf(); }

    void make_root() {
        assert(parent()->is_root());
        set_parent(parent()->parent());
    }

    // Getters for the key/value at position i in the node.
    const key_type &key(int i) const { return params_type::key(slot(i)); }

    reference value(int i) { return params_type::element(slot(i)); }

    const_reference value(int i) const { return params_type::element(slot(i)); }

    // Getters/setter for the child at position i in the node.
    btree_node *child(int i) const { return GetField<3>()[i]; }

    btree_node *&mutable_child(int i) { return GetField<3>()[i]; }

    void clear_child(int i) {
        abel::container_internal::sanitizer_poison_object(&mutable_child(i));
    }

    void set_child(int i, btree_node *c) {
        abel::container_internal::sanitizer_unpoison_object(&mutable_child(i));
        mutable_child(i) = c;
        c->set_position(i);
    }

    void init_child(int i, btree_node *c) {
        set_child(i, c);
        c->set_parent(this);
    }

    // Returns the position of the first value whose key is not less than k.
    template<typename K>
    SearchResult<int, is_key_compare_to::value> lower_bound(
            const K &k, const key_compare &comp) const {
        return use_linear_search::value ? linear_search(k, comp)
                                        : binary_search(k, comp);
    }

    // Returns the position of the first value whose key is greater than k.
    template<typename K>
    int upper_bound(const K &k, const key_compare &comp) const {
        auto upper_compare = upper_bound_adapter<key_compare>(comp);
        return use_linear_search::value ? linear_search(k, upper_compare).value
                                        : binary_search(k, upper_compare).value;
    }

    template<typename K, typename Compare>
    SearchResult<int, btree_is_key_compare_to<Compare, key_type>::value>
    linear_search(const K &k, const Compare &comp) const {
        return linear_search_impl(k, 0, count(), comp,
                                  btree_is_key_compare_to<Compare, key_type>());
    }

    template<typename K, typename Compare>
    SearchResult<int, btree_is_key_compare_to<Compare, key_type>::value>
    binary_search(const K &k, const Compare &comp) const {
        return binary_search_impl(k, 0, count(), comp,
                                  btree_is_key_compare_to<Compare, key_type>());
    }

    // Returns the position of the first value whose key is not less than k using
    // linear search performed using plain compare.
    template<typename K, typename Compare>
    SearchResult<int, false> linear_search_impl(
            const K &k, int s, const int e, const Compare &comp,
            std::false_type /* IsCompareTo */) const {
        while (s < e) {
            if (!comp(key(s), k)) {
                break;
            }
            ++s;
        }
        return {s};
    }

    // Returns the position of the first value whose key is not less than k using
    // linear search performed using compare-to.
    template<typename K, typename Compare>
    SearchResult<int, true> linear_search_impl(
            const K &k, int s, const int e, const Compare &comp,
            std::true_type /* IsCompareTo */) const {
        while (s < e) {
            const abel::weak_ordering c = comp(key(s), k);
            if (c == 0) {
                return {s, MatchKind::kEq};
            } else if (c > 0) {
                break;
            }
            ++s;
        }
        return {s, MatchKind::kNe};
    }

    // Returns the position of the first value whose key is not less than k using
    // binary search performed using plain compare.
    template<typename K, typename Compare>
    SearchResult<int, false> binary_search_impl(
            const K &k, int s, int e, const Compare &comp,
            std::false_type /* IsCompareTo */) const {
        while (s != e) {
            const int mid = (s + e) >> 1;
            if (comp(key(mid), k)) {
                s = mid + 1;
            } else {
                e = mid;
            }
        }
        return {s};
    }

    // Returns the position of the first value whose key is not less than k using
    // binary search performed using compare-to.
    template<typename K, typename CompareTo>
    SearchResult<int, true> binary_search_impl(
            const K &k, int s, int e, const CompareTo &comp,
            std::true_type /* IsCompareTo */) const {
        if (is_multi_container::value) {
            MatchKind exact_match = MatchKind::kNe;
            while (s != e) {
                const int mid = (s + e) >> 1;
                const abel::weak_ordering c = comp(key(mid), k);
                if (c < 0) {
                    s = mid + 1;
                } else {
                    e = mid;
                    if (c == 0) {
                        // Need to return the first value whose key is not less than k,
                        // which requires continuing the binary search if this is a
                        // multi-container.
                        exact_match = MatchKind::kEq;
                    }
                }
            }
            return {s, exact_match};
        } else {  // Not a multi-container.
            while (s != e) {
                const int mid = (s + e) >> 1;
                const abel::weak_ordering c = comp(key(mid), k);
                if (c < 0) {
                    s = mid + 1;
                } else if (c > 0) {
                    e = mid;
                } else {
                    return {mid, MatchKind::kEq};
                }
            }
            return {s, MatchKind::kNe};
        }
    }

    // Emplaces a value at position i, shifting all existing values and
    // children at positions >= i to the right by 1.
    template<typename... Args>
    void emplace_value(size_type i, allocator_type *alloc, Args &&... args);

    // Removes the value at position i, shifting all existing values and children
    // at positions > i to the left by 1.
    void remove_value(int i, allocator_type *alloc);

    // Removes the values at positions [i, i + to_erase), shifting all values
    // after that range to the left by to_erase. Does not change children at all.
    void remove_values_ignore_children(int i, int to_erase,
                                       allocator_type *alloc);

    // Rebalances a node with its right sibling.
    void rebalance_right_to_left(int to_move, btree_node *right,
                                 allocator_type *alloc);

    void rebalance_left_to_right(int to_move, btree_node *right,
                                 allocator_type *alloc);

    // Splits a node, moving a portion of the node's values to its right sibling.
    void split(int insert_position, btree_node *dest, allocator_type *alloc);

    // Merges a node with its right sibling, moving all of the values and the
    // delimiting key in the parent node onto itself.
    void merge(btree_node *sibling, allocator_type *alloc);

    // Swap the contents of "this" and "src".
    void swap(btree_node *src, allocator_type *alloc);

    // Node allocation/deletion routines.
    static btree_node *init_leaf(btree_node *n, btree_node *parent,
                                 int max_count) {
        n->set_parent(parent);
        n->set_position(0);
        n->set_start(0);
        n->set_count(0);
        n->set_max_count(max_count);
        abel::container_internal::sanitizer_poison_memory_region(
                n->slot(0), max_count * sizeof(slot_type));
        return n;
    }

    static btree_node *init_internal(btree_node *n, btree_node *parent) {
        init_leaf(n, parent, kNodeValues);
        // Set `max_count` to a sentinel value to indicate that this node is
        // internal.
        n->set_max_count(kInternalNodeMaxCount);
        abel::container_internal::sanitizer_poison_memory_region(
                &n->mutable_child(0), (kNodeValues + 1) * sizeof(btree_node *));
        return n;
    }

    void destroy(allocator_type *alloc) {
        for (int i = 0; i < count(); ++i) {
            value_destroy(i, alloc);
        }
    }

  public:
    // Exposed only for tests.
    static bool testonly_uses_linear_node_search() {
        return use_linear_search::value;
    }

  private:
    template<typename... Args>
    void value_init(const size_type i, allocator_type *alloc, Args &&... args) {
        abel::container_internal::sanitizer_unpoison_object(slot(i));
        params_type::construct(alloc, slot(i), std::forward<Args>(args)...);
    }

    void value_destroy(const size_type i, allocator_type *alloc) {
        params_type::destroy(alloc, slot(i));
        abel::container_internal::sanitizer_poison_object(slot(i));
    }

    // Move n values starting at value i in this node into the values starting at
    // value j in node x.
    void uninitialized_move_n(const size_type n, const size_type i,
                              const size_type j, btree_node *x,
                              allocator_type *alloc) {
        abel::container_internal::SanitizerUnpoisonMemoryRegion(
                x->slot(j), n * sizeof(slot_type));
        for (slot_type *src = slot(i), *end = src + n, *dest = x->slot(j);
             src != end; ++src, ++dest) {
            params_type::construct(alloc, dest, src);
        }
    }

    // Destroys a range of n values, starting at index i.
    void value_destroy_n(const size_type i, const size_type n,
                         allocator_type *alloc) {
        for (int j = 0; j < n; ++j) {
            value_destroy(i + j, alloc);
        }
    }

    template<typename P>
    friend
    class btree;

    template<typename N, typename R, typename P>
    friend
    struct btree_iterator;

    friend class BtreeNodePeer;
};

template<typename Node, typename Reference, typename Pointer>
struct btree_iterator {
  private:
    using key_type = typename Node::key_type;
    using size_type = typename Node::size_type;
    using params_type = typename Node::params_type;

    using node_type = Node;
    using normal_node = typename std::remove_const<Node>::type;
    using const_node = const Node;
    using normal_pointer = typename params_type::pointer;
    using normal_reference = typename params_type::reference;
    using const_pointer = typename params_type::const_pointer;
    using const_reference = typename params_type::const_reference;
    using slot_type = typename params_type::slot_type;

    using iterator =
    btree_iterator<normal_node, normal_reference, normal_pointer>;
    using const_iterator =
    btree_iterator<const_node, const_reference, const_pointer>;

  public:
    // These aliases are public for std::iterator_traits.
    using difference_type = typename Node::difference_type;
    using value_type = typename params_type::value_type;
    using pointer = Pointer;
    using reference = Reference;
    using iterator_category = std::bidirectional_iterator_tag;

    btree_iterator() : node(nullptr), position(-1) {}

    btree_iterator(Node *n, int p) : node(n), position(p) {}

    // NOTE: this SFINAE allows for implicit conversions from iterator to
    // const_iterator, but it specifically avoids defining copy constructors so
    // that btree_iterator can be trivially copyable. This is for performance and
    // binary size reasons.
    template<typename N, typename R, typename P,
            abel::enable_if_t<
                    std::is_same<btree_iterator<N, R, P>, iterator>::value &&
                    std::is_same<btree_iterator, const_iterator>::value,
                    int> = 0>
    btree_iterator(const btree_iterator<N, R, P> &x)  // NOLINT
            : node(x.node), position(x.position) {}

  private:
    // This SFINAE allows explicit conversions from const_iterator to
    // iterator, but also avoids defining a copy constructor.
    // NOTE: the const_cast is safe because this constructor is only called by
    // non-const methods and the container owns the nodes.
    template<typename N, typename R, typename P,
            abel::enable_if_t<
                    std::is_same<btree_iterator<N, R, P>, const_iterator>::value &&
                    std::is_same<btree_iterator, iterator>::value,
                    int> = 0>
    explicit btree_iterator(const btree_iterator<N, R, P> &x)
            : node(const_cast<node_type *>(x.node)), position(x.position) {}

    // Increment/decrement the iterator.
    void increment() {
        if (node->leaf() && ++position < node->count()) {
            return;
        }
        increment_slow();
    }

    void increment_slow();

    void decrement() {
        if (node->leaf() && --position >= 0) {
            return;
        }
        decrement_slow();
    }

    void decrement_slow();

  public:
    bool operator==(const const_iterator &x) const {
        return node == x.node && position == x.position;
    }

    bool operator!=(const const_iterator &x) const {
        return node != x.node || position != x.position;
    }

    // Accessors for the key/value the iterator is pointing at.
    reference operator*() const {
        return node->value(position);
    }

    pointer operator->() const {
        return &node->value(position);
    }

    btree_iterator &operator++() {
        increment();
        return *this;
    }

    btree_iterator &operator--() {
        decrement();
        return *this;
    }

    btree_iterator operator++(int) {
        btree_iterator tmp = *this;
        ++*this;
        return tmp;
    }

    btree_iterator operator--(int) {
        btree_iterator tmp = *this;
        --*this;
        return tmp;
    }

  private:
    template<typename Params>
    friend
    class btree;

    template<typename Tree>
    friend
    class btree_container;

    template<typename Tree>
    friend
    class btree_set_container;

    template<typename Tree>
    friend
    class btree_map_container;

    template<typename Tree>
    friend
    class btree_multiset_container;

    template<typename N, typename R, typename P>
    friend
    struct btree_iterator;

    template<typename TreeType, typename CheckerType>
    friend
    class base_checker;

    const key_type &key() const { return node->key(position); }

    slot_type *slot() { return node->slot(position); }

    // The node in the tree the iterator is pointing at.
    Node *node;
    // The position within the node of the tree the iterator is pointing at.
    // TODO(ezb): make this a field_type
    int position;
};

template<typename Params>
class btree {
    using node_type = btree_node<Params>;
    using is_key_compare_to = typename Params::is_key_compare_to;

    // We use a static empty node for the root/leftmost/rightmost of empty btrees
    // in order to avoid branching in begin()/end().
    struct alignas(node_type::Alignment()) EmptyNodeType : node_type {
        using field_type = typename node_type::field_type;
        node_type *parent;
        field_type position = 0;
        field_type start = 0;
        field_type count = 0;
        // max_count must be != kInternalNodeMaxCount (so that this node is regarded
        // as a leaf node). max_count() is never called when the tree is empty.
        field_type max_count = node_type::kInternalNodeMaxCount + 1;

#ifdef _MSC_VER
        // MSVC has constexpr code generations bugs here.
        EmptyNodeType() : parent(this) {}
#else

        constexpr EmptyNodeType(node_type *p) : parent(p) {}

#endif
    };

    static node_type *EmptyNode() {
#ifdef _MSC_VER
        static EmptyNodeType* empty_node = new EmptyNodeType;
        // This assert fails on some other construction methods.
        assert(empty_node->parent == empty_node);
        return empty_node;
#else
        static constexpr EmptyNodeType empty_node(
                const_cast<EmptyNodeType *>(&empty_node));
        return const_cast<EmptyNodeType *>(&empty_node);
#endif
    }

    enum {
        kNodeValues = node_type::kNodeValues,
        kMinNodeValues = kNodeValues / 2,
    };

    struct node_stats {
        using size_type = typename Params::size_type;

        node_stats(size_type l, size_type i)
                : leaf_nodes(l),
                  internal_nodes(i) {
        }

        node_stats &operator+=(const node_stats &x) {
            leaf_nodes += x.leaf_nodes;
            internal_nodes += x.internal_nodes;
            return *this;
        }

        size_type leaf_nodes;
        size_type internal_nodes;
    };

  public:
    using key_type = typename Params::key_type;
    using value_type = typename Params::value_type;
    using size_type = typename Params::size_type;
    using difference_type = typename Params::difference_type;
    using key_compare = typename Params::key_compare;
    using value_compare = typename Params::value_compare;
    using allocator_type = typename Params::allocator_type;
    using reference = typename Params::reference;
    using const_reference = typename Params::const_reference;
    using pointer = typename Params::pointer;
    using const_pointer = typename Params::const_pointer;
    using iterator = btree_iterator<node_type, reference, pointer>;
    using const_iterator = typename iterator::const_iterator;
    using reverse_iterator = std::reverse_iterator<iterator>;
    using const_reverse_iterator = std::reverse_iterator<const_iterator>;
    using node_handle_type = node_handle<Params, Params, allocator_type>;

    // Internal types made public for use by btree_container types.
    using params_type = Params;
    using slot_type = typename Params::slot_type;

  private:
    // For use in copy_or_move_values_in_order.
    const value_type &maybe_move_from_iterator(const_iterator x) { return *x; }

    value_type &&maybe_move_from_iterator(iterator x) { return std::move(*x); }

    // Copies or moves (depending on the template parameter) the values in
    // x into this btree in their order in x. This btree must be empty before this
    // method is called. This method is used in copy construction, copy
    // assignment, and move assignment.
    template<typename Btree>
    void copy_or_move_values_in_order(Btree *x);

    // Validates that various assumptions/requirements are true at compile time.
    constexpr static bool static_assert_validation();

  public:
    btree(const key_compare &comp, const allocator_type &alloc);

    btree(const btree &x);

    btree(btree &&x) noexcept
            : root_(std::move(x.root_)),
              rightmost_(abel::exchange(x.rightmost_, EmptyNode())),
              size_(abel::exchange(x.size_, 0)) {
        x.mutable_root() = EmptyNode();
    }

    ~btree() {
        // Put static_asserts in destructor to avoid triggering them before the type
        // is complete.
        static_assert(static_assert_validation(), "This call must be elided.");
        clear();
    }

    // Assign the contents of x to *this.
    btree &operator=(const btree &x);

    btree &operator=(btree &&x) noexcept;

    iterator begin() {
        return iterator(leftmost(), 0);
    }

    const_iterator begin() const {
        return const_iterator(leftmost(), 0);
    }

    iterator end() { return iterator(rightmost_, rightmost_->count()); }

    const_iterator end() const {
        return const_iterator(rightmost_, rightmost_->count());
    }

    reverse_iterator rbegin() {
        return reverse_iterator(end());
    }

    const_reverse_iterator rbegin() const {
        return const_reverse_iterator(end());
    }

    reverse_iterator rend() {
        return reverse_iterator(begin());
    }

    const_reverse_iterator rend() const {
        return const_reverse_iterator(begin());
    }

    // Finds the first element whose key is not less than key.
    template<typename K>
    iterator lower_bound(const K &key) {
        return internal_end(internal_lower_bound(key));
    }

    template<typename K>
    const_iterator lower_bound(const K &key) const {
        return internal_end(internal_lower_bound(key));
    }

    // Finds the first element whose key is greater than key.
    template<typename K>
    iterator upper_bound(const K &key) {
        return internal_end(internal_upper_bound(key));
    }

    template<typename K>
    const_iterator upper_bound(const K &key) const {
        return internal_end(internal_upper_bound(key));
    }

    // Finds the range of values which compare equal to key. The first member of
    // the returned pair is equal to lower_bound(key). The second member pair of
    // the pair is equal to upper_bound(key).
    template<typename K>
    std::pair<iterator, iterator> equal_range(const K &key) {
        return {lower_bound(key), upper_bound(key)};
    }

    template<typename K>
    std::pair<const_iterator, const_iterator> equal_range(const K &key) const {
        return {lower_bound(key), upper_bound(key)};
    }

    // Inserts a value into the btree only if it does not already exist. The
    // boolean return value indicates whether insertion succeeded or failed.
    // Requirement: if `key` already exists in the btree, does not consume `args`.
    // Requirement: `key` is never referenced after consuming `args`.
    template<typename... Args>
    std::pair<iterator, bool> insert_unique(const key_type &key, Args &&... args);

    // Inserts with hint. Checks to see if the value should be placed immediately
    // before `position` in the tree. If so, then the insertion will take
    // amortized constant time. If not, the insertion will take amortized
    // logarithmic time as if a call to insert_unique() were made.
    // Requirement: if `key` already exists in the btree, does not consume `args`.
    // Requirement: `key` is never referenced after consuming `args`.
    template<typename... Args>
    std::pair<iterator, bool> insert_hint_unique(iterator position,
                                                 const key_type &key,
                                                 Args &&... args);

    // Insert a range of values into the btree.
    template<typename InputIterator>
    void insert_iterator_unique(InputIterator b, InputIterator e);

    // Inserts a value into the btree.
    template<typename ValueType>
    iterator insert_multi(const key_type &key, ValueType &&v);

    // Inserts a value into the btree.
    template<typename ValueType>
    iterator insert_multi(ValueType &&v) {
        return insert_multi(params_type::key(v), std::forward<ValueType>(v));
    }

    // Insert with hint. Check to see if the value should be placed immediately
    // before position in the tree. If it does, then the insertion will take
    // amortized constant time. If not, the insertion will take amortized
    // logarithmic time as if a call to insert_multi(v) were made.
    template<typename ValueType>
    iterator insert_hint_multi(iterator position, ValueType &&v);

    // Insert a range of values into the btree.
    template<typename InputIterator>
    void insert_iterator_multi(InputIterator b, InputIterator e);

    // Erase the specified iterator from the btree. The iterator must be valid
    // (i.e. not equal to end()).  Return an iterator pointing to the node after
    // the one that was erased (or end() if none exists).
    // Requirement: does not read the value at `*iter`.
    iterator erase(iterator iter);

    // Erases range. Returns the number of keys erased and an iterator pointing
    // to the element after the last erased element.
    std::pair<size_type, iterator> erase(iterator begin, iterator end);

    // Erases the specified key from the btree. Returns 1 if an element was
    // erased and 0 otherwise.
    template<typename K>
    size_type erase_unique(const K &key);

    // Erases all of the entries matching the specified key from the
    // btree. Returns the number of elements erased.
    template<typename K>
    size_type erase_multi(const K &key);

    // Finds the iterator corresponding to a key or returns end() if the key is
    // not present.
    template<typename K>
    iterator find(const K &key) {
        return internal_end(internal_find(key));
    }

    template<typename K>
    const_iterator find(const K &key) const {
        return internal_end(internal_find(key));
    }

    // Returns a count of the number of times the key appears in the btree.
    template<typename K>
    size_type count_unique(const K &key) const {
        const iterator begin = internal_find(key);
        if (begin.node == nullptr) {
            // The key doesn't exist in the tree.
            return 0;
        }
        return 1;
    }

    // Returns a count of the number of times the key appears in the btree.
    template<typename K>
    size_type count_multi(const K &key) const {
        const auto range = equal_range(key);
        return std::distance(range.first, range.second);
    }

    // Clear the btree, deleting all of the values it contains.
    void clear();

    // Swap the contents of *this and x.
    void swap(btree &x);

    const key_compare &key_comp() const noexcept {
        return root_.template get<0>();
    }

    template<typename K, typename LK>
    bool compare_keys(const K &x, const LK &y) const {
        return compare_internal::compare_result_as_less_than(key_comp()(x, y));
    }

    value_compare value_comp() const { return value_compare(key_comp()); }

    // Verifies the structure of the btree.
    void verify() const;

    // Size routines.
    size_type size() const { return size_; }

    size_type max_size() const { return (std::numeric_limits<size_type>::max)(); }

    bool empty() const { return size_ == 0; }

    // The height of the btree. An empty tree will have height 0.
    size_type height() const {
        size_type h = 0;
        if (!empty()) {
            // Count the length of the chain from the leftmost node up to the
            // root. We actually count from the root back around to the level below
            // the root, but the calculation is the same because of the circularity
            // of that traversal.
            const node_type *n = root();
            do {
                ++h;
                n = n->parent();
            } while (n != root());
        }
        return h;
    }

    // The number of internal, leaf and total nodes used by the btree.
    size_type leaf_nodes() const {
        return internal_stats(root()).leaf_nodes;
    }

    size_type internal_nodes() const {
        return internal_stats(root()).internal_nodes;
    }

    size_type nodes() const {
        node_stats stats = internal_stats(root());
        return stats.leaf_nodes + stats.internal_nodes;
    }

    // The total number of bytes used by the btree.
    size_type bytes_used() const {
        node_stats stats = internal_stats(root());
        if (stats.leaf_nodes == 1 && stats.internal_nodes == 0) {
            return sizeof(*this) +
                   node_type::LeafSize(root()->max_count());
        } else {
            return sizeof(*this) +
                   stats.leaf_nodes * node_type::LeafSize() +
                   stats.internal_nodes * node_type::InternalSize();
        }
    }

    // The average number of bytes used per value stored in the btree.
    static double average_bytes_per_value() {
        // Returns the number of bytes per value on a leaf node that is 75%
        // full. Experimentally, this matches up nicely with the computed number of
        // bytes per value in trees that had their values inserted in random order.
        return node_type::LeafSize() / (kNodeValues * 0.75);
    }

    // The fullness of the btree. Computed as the number of elements in the btree
    // divided by the maximum number of elements a tree with the current number
    // of nodes could hold. A value of 1 indicates perfect space
    // utilization. Smaller values indicate space wastage.
    // Returns 0 for empty trees.
    double fullness() const {
        if (empty()) return 0.0;
        return static_cast<double>(size()) / (nodes() * kNodeValues);
    }

    // The overhead of the btree structure in bytes per node. Computed as the
    // total number of bytes used by the btree minus the number of bytes used for
    // storing elements divided by the number of elements.
    // Returns 0 for empty trees.
    double overhead() const {
        if (empty()) return 0.0;
        return (bytes_used() - size() * sizeof(value_type)) /
               static_cast<double>(size());
    }

    // The allocator used by the btree.
    allocator_type get_allocator() const {
        return allocator();
    }

  private:
    // Internal accessor routines.
    node_type *root() { return root_.template get<2>(); }

    const node_type *root() const { return root_.template get<2>(); }

    node_type *&mutable_root() noexcept { return root_.template get<2>(); }

    key_compare *mutable_key_comp() noexcept { return &root_.template get<0>(); }

    // The leftmost node is stored as the parent of the root node.
    node_type *leftmost() { return root()->parent(); }

    const node_type *leftmost() const { return root()->parent(); }

    // Allocator routines.
    allocator_type *mutable_allocator() noexcept {
        return &root_.template get<1>();
    }

    const allocator_type &allocator() const noexcept {
        return root_.template get<1>();
    }

    // Allocates a correctly aligned node of at least size bytes using the
    // allocator.
    node_type *allocate(const size_type size) {
        return reinterpret_cast<node_type *>(
                abel::container_internal::Allocate<node_type::Alignment()>(
                        mutable_allocator(), size));
    }

    // Node creation/deletion routines.
    node_type *new_internal_node(node_type *parent) {
        node_type *p = allocate(node_type::InternalSize());
        return node_type::init_internal(p, parent);
    }

    node_type *new_leaf_node(node_type *parent) {
        node_type *p = allocate(node_type::LeafSize());
        return node_type::init_leaf(p, parent, kNodeValues);
    }

    node_type *new_leaf_root_node(const int max_count) {
        node_type *p = allocate(node_type::LeafSize(max_count));
        return node_type::init_leaf(p, p, max_count);
    }

    // Deletion helper routines.
    void erase_same_node(iterator begin, iterator end);

    iterator erase_from_leaf_node(iterator begin, size_type to_erase);

    iterator rebalance_after_delete(iterator iter);

    // Deallocates a node of a certain size in bytes using the allocator.
    void deallocate(const size_type size, node_type *node) {
        abel::container_internal::Deallocate<node_type::Alignment()>(
                mutable_allocator(), node, size);
    }

    void delete_internal_node(node_type *node) {
        node->destroy(mutable_allocator());
        deallocate(node_type::InternalSize(), node);
    }

    void delete_leaf_node(node_type *node) {
        node->destroy(mutable_allocator());
        deallocate(node_type::LeafSize(node->max_count()), node);
    }

    // Rebalances or splits the node iter points to.
    void rebalance_or_split(iterator *iter);

    // Merges the values of left, right and the delimiting key on their parent
    // onto left, removing the delimiting key and deleting right.
    void merge_nodes(node_type *left, node_type *right);

    // Tries to merge node with its left or right sibling, and failing that,
    // rebalance with its left or right sibling. Returns true if a merge
    // occurred, at which point it is no longer valid to access node. Returns
    // false if no merging took place.
    bool try_merge_or_rebalance(iterator *iter);

    // Tries to shrink the height of the tree by 1.
    void try_shrink();

    iterator internal_end(iterator iter) {
        return iter.node != nullptr ? iter : end();
    }

    const_iterator internal_end(const_iterator iter) const {
        return iter.node != nullptr ? iter : end();
    }

    // Emplaces a value into the btree immediately before iter. Requires that
    // key(v) <= iter.key() and (--iter).key() <= key(v).
    template<typename... Args>
    iterator internal_emplace(iterator iter, Args &&... args);

    // Returns an iterator pointing to the first value >= the value "iter" is
    // pointing at. Note that "iter" might be pointing to an invalid location as
    // iter.position == iter.node->count(). This routine simply moves iter up in
    // the tree to a valid location.
    // Requires: iter.node is non-null.
    template<typename IterType>
    static IterType internal_last(IterType iter);

    // Returns an iterator pointing to the leaf position at which key would
    // reside in the tree. We provide 2 versions of internal_locate. The first
    // version uses a less-than comparator and is incapable of distinguishing when
    // there is an exact match. The second version is for the key-compare-to
    // specialization and distinguishes exact matches. The key-compare-to
    // specialization allows the caller to avoid a subsequent comparison to
    // determine if an exact match was made, which is important for keys with
    // expensive comparison, such as strings.
    template<typename K>
    SearchResult<iterator, is_key_compare_to::value> internal_locate(
            const K &key) const;

    template<typename K>
    SearchResult<iterator, false> internal_locate_impl(
            const K &key, std::false_type /* IsCompareTo */) const;

    template<typename K>
    SearchResult<iterator, true> internal_locate_impl(
            const K &key, std::true_type /* IsCompareTo */) const;

    // Internal routine which implements lower_bound().
    template<typename K>
    iterator internal_lower_bound(const K &key) const;

    // Internal routine which implements upper_bound().
    template<typename K>
    iterator internal_upper_bound(const K &key) const;

    // Internal routine which implements find().
    template<typename K>
    iterator internal_find(const K &key) const;

    // Deletes a node and all of its children.
    void internal_clear(node_type *node);

    // Verifies the tree structure of node.
    int internal_verify(const node_type *node,
                        const key_type *lo, const key_type *hi) const;

    node_stats internal_stats(const node_type *node) const {
        // The root can be a static empty node.
        if (node == nullptr || (node == root() && empty())) {
            return node_stats(0, 0);
        }
        if (node->leaf()) {
            return node_stats(1, 0);
        }
        node_stats res(0, 1);
        for (int i = 0; i <= node->count(); ++i) {
            res += internal_stats(node->child(i));
        }
        return res;
    }

  public:
    // Exposed only for tests.
    static bool testonly_uses_linear_node_search() {
        return node_type::testonly_uses_linear_node_search();
    }

  private:
    // We use compressed tuple in order to save space because key_compare and
    // allocator_type are usually empty.
    abel::container_internal::compressed_tuple<key_compare, allocator_type,
            node_type *>
            root_;

    // A pointer to the rightmost node. Note that the leftmost node is stored as
    // the root's parent.
    node_type *rightmost_;

    // Number of values.
    size_type size_;
};

////
// btree_node methods
template<typename P>
template<typename... Args>
ABEL_FORCE_INLINE void btree_node<P>::emplace_value(const size_type i,
                                                    allocator_type *alloc,
                                                    Args &&... args) {
    assert(i <= count());
    // Shift old values to create space for new value and then construct it in
    // place.
    if (i < count()) {
        value_init(count(), alloc, slot(count() - 1));
        for (size_type j = count() - 1; j > i; --j)
            params_type::move(alloc, slot(j - 1), slot(j));
        value_destroy(i, alloc);
    }
    value_init(i, alloc, std::forward<Args>(args)...);
    set_count(count() + 1);

    if (!leaf() && count() > i + 1) {
        for (int j = count(); j > i + 1; --j) {
            set_child(j, child(j - 1));
        }
        clear_child(i + 1);
    }
}

template<typename P>
ABEL_FORCE_INLINE void btree_node<P>::remove_value(const int i, allocator_type *alloc) {
    if (!leaf() && count() > i + 1) {
        assert(child(i + 1)->count() == 0);
        for (size_type j = i + 1; j < count(); ++j) {
            set_child(j, child(j + 1));
        }
        clear_child(count());
    }

    remove_values_ignore_children(i, /*to_erase=*/1, alloc);
}

template<typename P>
ABEL_FORCE_INLINE void btree_node<P>::remove_values_ignore_children(
        const int i, const int to_erase, allocator_type *alloc) {
    params_type::move(alloc, slot(i + to_erase), slot(count()), slot(i));
    value_destroy_n(count() - to_erase, to_erase, alloc);
    set_count(count() - to_erase);
}

template<typename P>
void btree_node<P>::rebalance_right_to_left(const int to_move,
                                            btree_node *right,
                                            allocator_type *alloc) {
    assert(parent() == right->parent());
    assert(position() + 1 == right->position());
    assert(right->count() >= count());
    assert(to_move >= 1);
    assert(to_move <= right->count());

    // 1) Move the delimiting value in the parent to the left node.
    value_init(count(), alloc, parent()->slot(position()));

    // 2) Move the (to_move - 1) values from the right node to the left node.
    right->uninitialized_move_n(to_move - 1, 0, count() + 1, this, alloc);

    // 3) Move the new delimiting value to the parent from the right node.
    params_type::move(alloc, right->slot(to_move - 1),
                      parent()->slot(position()));

    // 4) Shift the values in the right node to their correct position.
    params_type::move(alloc, right->slot(to_move), right->slot(right->count()),
                      right->slot(0));

    // 5) Destroy the now-empty to_move entries in the right node.
    right->value_destroy_n(right->count() - to_move, to_move, alloc);

    if (!leaf()) {
        // Move the child pointers from the right to the left node.
        for (int i = 0; i < to_move; ++i) {
            init_child(count() + i + 1, right->child(i));
        }
        for (int i = 0; i <= right->count() - to_move; ++i) {
            assert(i + to_move <= right->max_count());
            right->init_child(i, right->child(i + to_move));
            right->clear_child(i + to_move);
        }
    }

    // Fixup the counts on the left and right nodes.
    set_count(count() + to_move);
    right->set_count(right->count() - to_move);
}

template<typename P>
void btree_node<P>::rebalance_left_to_right(const int to_move,
                                            btree_node *right,
                                            allocator_type *alloc) {
    assert(parent() == right->parent());
    assert(position() + 1 == right->position());
    assert(count() >= right->count());
    assert(to_move >= 1);
    assert(to_move <= count());

    // Values in the right node are shifted to the right to make room for the
    // new to_move values. Then, the delimiting value in the parent and the
    // other (to_move - 1) values in the left node are moved into the right node.
    // Lastly, a new delimiting value is moved from the left node into the
    // parent, and the remaining empty left node entries are destroyed.

    if (right->count() >= to_move) {
        // The original location of the right->count() values are sufficient to hold
        // the new to_move entries from the parent and left node.

        // 1) Shift existing values in the right node to their correct positions.
        right->uninitialized_move_n(to_move, right->count() - to_move,
                                    right->count(), right, alloc);
        for (slot_type *src = right->slot(right->count() - to_move - 1),
                     *dest = right->slot(right->count() - 1),
                     *end = right->slot(0);
             src >= end; --src, --dest) {
            params_type::move(alloc, src, dest);
        }

        // 2) Move the delimiting value in the parent to the right node.
        params_type::move(alloc, parent()->slot(position()),
                          right->slot(to_move - 1));

        // 3) Move the (to_move - 1) values from the left node to the right node.
        params_type::move(alloc, slot(count() - (to_move - 1)), slot(count()),
                          right->slot(0));
    } else {
        // The right node does not have enough initialized space to hold the new
        // to_move entries, so part of them will move to uninitialized space.

        // 1) Shift existing values in the right node to their correct positions.
        right->uninitialized_move_n(right->count(), 0, to_move, right, alloc);

        // 2) Move the delimiting value in the parent to the right node.
        right->value_init(to_move - 1, alloc, parent()->slot(position()));

        // 3) Move the (to_move - 1) values from the left node to the right node.
        const size_type uninitialized_remaining = to_move - right->count() - 1;
        uninitialized_move_n(uninitialized_remaining,
                             count() - uninitialized_remaining, right->count(),
                             right, alloc);
        params_type::move(alloc, slot(count() - (to_move - 1)),
                          slot(count() - uninitialized_remaining), right->slot(0));
    }

    // 4) Move the new delimiting value to the parent from the left node.
    params_type::move(alloc, slot(count() - to_move), parent()->slot(position()));

    // 5) Destroy the now-empty to_move entries in the left node.
    value_destroy_n(count() - to_move, to_move, alloc);

    if (!leaf()) {
        // Move the child pointers from the left to the right node.
        for (int i = right->count(); i >= 0; --i) {
            right->init_child(i + to_move, right->child(i));
            right->clear_child(i);
        }
        for (int i = 1; i <= to_move; ++i) {
            right->init_child(i - 1, child(count() - to_move + i));
            clear_child(count() - to_move + i);
        }
    }

    // Fixup the counts on the left and right nodes.
    set_count(count() - to_move);
    right->set_count(right->count() + to_move);
}

template<typename P>
void btree_node<P>::split(const int insert_position, btree_node *dest,
                          allocator_type *alloc) {
    assert(dest->count() == 0);
    assert(max_count() == kNodeValues);

    // We bias the split based on the position being inserted. If we're
    // inserting at the beginning of the left node then bias the split to put
    // more values on the right node. If we're inserting at the end of the
    // right node then bias the split to put more values on the left node.
    if (insert_position == 0) {
        dest->set_count(count() - 1);
    } else if (insert_position == kNodeValues) {
        dest->set_count(0);
    } else {
        dest->set_count(count() / 2);
    }
    set_count(count() - dest->count());
    assert(count() >= 1);

    // Move values from the left sibling to the right sibling.
    uninitialized_move_n(dest->count(), count(), 0, dest, alloc);

    // Destroy the now-empty entries in the left node.
    value_destroy_n(count(), dest->count(), alloc);

    // The split key is the largest value in the left sibling.
    set_count(count() - 1);
    parent()->emplace_value(position(), alloc, slot(count()));
    value_destroy(count(), alloc);
    parent()->init_child(position() + 1, dest);

    if (!leaf()) {
        for (int i = 0; i <= dest->count(); ++i) {
            assert(child(count() + i + 1) != nullptr);
            dest->init_child(i, child(count() + i + 1));
            clear_child(count() + i + 1);
        }
    }
}

template<typename P>
void btree_node<P>::merge(btree_node *src, allocator_type *alloc) {
    assert(parent() == src->parent());
    assert(position() + 1 == src->position());

    // Move the delimiting value to the left node.
    value_init(count(), alloc, parent()->slot(position()));

    // Move the values from the right to the left node.
    src->uninitialized_move_n(src->count(), 0, count() + 1, this, alloc);

    // Destroy the now-empty entries in the right node.
    src->value_destroy_n(0, src->count(), alloc);

    if (!leaf()) {
        // Move the child pointers from the right to the left node.
        for (int i = 0; i <= src->count(); ++i) {
            init_child(count() + i + 1, src->child(i));
            src->clear_child(i);
        }
    }

    // Fixup the counts on the src and dest nodes.
    set_count(1 + count() + src->count());
    src->set_count(0);

    // Remove the value on the parent node.
    parent()->remove_value(position(), alloc);
}

template<typename P>
void btree_node<P>::swap(btree_node *x, allocator_type *alloc) {
    using std::swap;
    assert(leaf() == x->leaf());

    // Determine which is the smaller/larger node.
    btree_node *smaller = this, *larger = x;
    if (smaller->count() > larger->count()) {
        swap(smaller, larger);
    }

    // Swap the values.
    for (slot_type *a = smaller->slot(0), *b = larger->slot(0),
                 *end = a + smaller->count();
         a != end; ++a, ++b) {
        params_type::swap(alloc, a, b);
    }

    // Move values that can't be swapped.
    const size_type to_move = larger->count() - smaller->count();
    larger->uninitialized_move_n(to_move, smaller->count(), smaller->count(),
                                 smaller, alloc);
    larger->value_destroy_n(smaller->count(), to_move, alloc);

    if (!leaf()) {
        // Swap the child pointers.
        std::swap_ranges(&smaller->mutable_child(0),
                         &smaller->mutable_child(smaller->count() + 1),
                         &larger->mutable_child(0));
        // Update swapped children's parent pointers.
        int i = 0;
        for (; i <= smaller->count(); ++i) {
            smaller->child(i)->set_parent(smaller);
            larger->child(i)->set_parent(larger);
        }
        // Move the child pointers that couldn't be swapped.
        for (; i <= larger->count(); ++i) {
            smaller->init_child(i, larger->child(i));
            larger->clear_child(i);
        }
    }

    // Swap the counts.
    swap(mutable_count(), x->mutable_count());
}

////
// btree_iterator methods
template<typename N, typename R, typename P>
void btree_iterator<N, R, P>::increment_slow() {
    if (node->leaf()) {
        assert(position >= node->count());
        btree_iterator save(*this);
        while (position == node->count() && !node->is_root()) {
            assert(node->parent()->child(node->position()) == node);
            position = node->position();
            node = node->parent();
        }
        if (position == node->count()) {
            *this = save;
        }
    } else {
        assert(position < node->count());
        node = node->child(position + 1);
        while (!node->leaf()) {
            node = node->child(0);
        }
        position = 0;
    }
}

template<typename N, typename R, typename P>
void btree_iterator<N, R, P>::decrement_slow() {
    if (node->leaf()) {
        assert(position <= -1);
        btree_iterator save(*this);
        while (position < 0 && !node->is_root()) {
            assert(node->parent()->child(node->position()) == node);
            position = node->position() - 1;
            node = node->parent();
        }
        if (position < 0) {
            *this = save;
        }
    } else {
        assert(position >= 0);
        node = node->child(position);
        while (!node->leaf()) {
            node = node->child(node->count());
        }
        position = node->count() - 1;
    }
}

////
// btree methods
template<typename P>
template<typename Btree>
void btree<P>::copy_or_move_values_in_order(Btree *x) {
    static_assert(std::is_same<btree, Btree>::value ||
                  std::is_same<const btree, Btree>::value,
                  "Btree type must be same or const.");
    assert(empty());

    // We can avoid key comparisons because we know the order of the
    // values is the same order we'll store them in.
    auto iter = x->begin();
    if (iter == x->end()) return;
    insert_multi(maybe_move_from_iterator(iter));
    ++iter;
    for (; iter != x->end(); ++iter) {
        // If the btree is not empty, we can just insert the new value at the end
        // of the tree.
        internal_emplace(end(), maybe_move_from_iterator(iter));
    }
}

template<typename P>
constexpr bool btree<P>::static_assert_validation() {
    static_assert(std::is_nothrow_copy_constructible<key_compare>::value,
                  "Key comparison must be nothrow copy constructible");
    static_assert(std::is_nothrow_copy_constructible<allocator_type>::value,
                  "Allocator must be nothrow copy constructible");
    static_assert(abel::is_trivially_copyable<iterator>::value,
                  "iterator not trivially copyable.");

    // Note: We assert that kTargetValues, which is computed from
    // Params::kTargetNodeSize, must fit the node_type::field_type.
    static_assert(
            kNodeValues < (1 << (8 * sizeof(typename node_type::field_type))),
            "target node size too large");

    // Verify that key_compare returns an abel::{weak,strong}_ordering or bool.
    using compare_result_type =
    abel::result_of_t<key_compare(key_type, key_type)>;
    static_assert(
            std::is_same<compare_result_type, bool>::value ||
            std::is_convertible<compare_result_type, abel::weak_ordering>::value,
            "key comparison function must return abel::{weak,strong}_ordering or "
            "bool.");

    // Test the assumption made in setting kNodeValueSpace.
    static_assert(node_type::MinimumOverhead() >= sizeof(void *) + 4,
                  "node space assumption incorrect");

    return true;
}

template<typename P>
btree<P>::btree(const key_compare &comp, const allocator_type &alloc)
        : root_(comp, alloc, EmptyNode()), rightmost_(EmptyNode()), size_(0) {}

template<typename P>
btree<P>::btree(const btree &x) : btree(x.key_comp(), x.allocator()) {
    copy_or_move_values_in_order(&x);
}

template<typename P>
template<typename... Args>
auto btree<P>::insert_unique(const key_type &key, Args &&... args)
-> std::pair<iterator, bool> {
    if (empty()) {
        mutable_root() = rightmost_ = new_leaf_root_node(1);
    }

    auto res = internal_locate(key);
    iterator &iter = res.value;

    if (res.HasMatch()) {
        if (res.IsEq()) {
            // The key already exists in the tree, do nothing.
            return {iter, false};
        }
    } else {
        iterator last = internal_last(iter);
        if (last.node && !compare_keys(key, last.key())) {
            // The key already exists in the tree, do nothing.
            return {last, false};
        }
    }
    return {internal_emplace(iter, std::forward<Args>(args)...), true};
}

template<typename P>
template<typename... Args>
ABEL_FORCE_INLINE auto btree<P>::insert_hint_unique(iterator position, const key_type &key,
                                                    Args &&... args)
-> std::pair<iterator, bool> {
    if (!empty()) {
        if (position == end() || compare_keys(key, position.key())) {
            iterator prev = position;
            if (position == begin() || compare_keys((--prev).key(), key)) {
                // prev.key() < key < position.key()
                return {internal_emplace(position, std::forward<Args>(args)...), true};
            }
        } else if (compare_keys(position.key(), key)) {
            ++position;
            if (position == end() || compare_keys(key, position.key())) {
                // {original `position`}.key() < key < {current `position`}.key()
                return {internal_emplace(position, std::forward<Args>(args)...), true};
            }
        } else {
            // position.key() == key
            return {position, false};
        }
    }
    return insert_unique(key, std::forward<Args>(args)...);
}

template<typename P>
template<typename InputIterator>
void btree<P>::insert_iterator_unique(InputIterator b, InputIterator e) {
    for (; b != e; ++b) {
        insert_hint_unique(end(), params_type::key(*b), *b);
    }
}

template<typename P>
template<typename ValueType>
auto btree<P>::insert_multi(const key_type &key, ValueType &&v) -> iterator {
    if (empty()) {
        mutable_root() = rightmost_ = new_leaf_root_node(1);
    }

    iterator iter = internal_upper_bound(key);
    if (iter.node == nullptr) {
        iter = end();
    }
    return internal_emplace(iter, std::forward<ValueType>(v));
}

template<typename P>
template<typename ValueType>
auto btree<P>::insert_hint_multi(iterator position, ValueType &&v) -> iterator {
    if (!empty()) {
        const key_type &key = params_type::key(v);
        if (position == end() || !compare_keys(position.key(), key)) {
            iterator prev = position;
            if (position == begin() || !compare_keys(key, (--prev).key())) {
                // prev.key() <= key <= position.key()
                return internal_emplace(position, std::forward<ValueType>(v));
            }
        } else {
            iterator next = position;
            ++next;
            if (next == end() || !compare_keys(next.key(), key)) {
                // position.key() < key <= next.key()
                return internal_emplace(next, std::forward<ValueType>(v));
            }
        }
    }
    return insert_multi(std::forward<ValueType>(v));
}

template<typename P>
template<typename InputIterator>
void btree<P>::insert_iterator_multi(InputIterator b, InputIterator e) {
    for (; b != e; ++b) {
        insert_hint_multi(end(), *b);
    }
}

template<typename P>
auto btree<P>::operator=(const btree &x) -> btree & {
    if (this != &x) {
        clear();

        *mutable_key_comp() = x.key_comp();
        if (abel::allocator_traits<
                allocator_type>::propagate_on_container_copy_assignment::value) {
            *mutable_allocator() = x.allocator();
        }

        copy_or_move_values_in_order(&x);
    }
    return *this;
}

template<typename P>
auto btree<P>::operator=(btree &&x) noexcept -> btree & {
    if (this != &x) {
        clear();

        using std::swap;
        if (abel::allocator_traits<
                allocator_type>::propagate_on_container_copy_assignment::value) {
            // Note: `root_` also contains the allocator and the key comparator.
            swap(root_, x.root_);
            swap(rightmost_, x.rightmost_);
            swap(size_, x.size_);
        } else {
            if (allocator() == x.allocator()) {
                swap(mutable_root(), x.mutable_root());
                swap(*mutable_key_comp(), *x.mutable_key_comp());
                swap(rightmost_, x.rightmost_);
                swap(size_, x.size_);
            } else {
                // We aren't allowed to propagate the allocator and the allocator is
                // different so we can't take over its memory. We must move each element
                // individually. We need both `x` and `this` to have `x`s key comparator
                // while moving the values so we can't swap the key comparators.
                *mutable_key_comp() = x.key_comp();
                copy_or_move_values_in_order(&x);
            }
        }
    }
    return *this;
}

template<typename P>
auto btree<P>::erase(iterator iter) -> iterator {
    bool internal_delete = false;
    if (!iter.node->leaf()) {
        // Deletion of a value on an internal node. First, move the largest value
        // from our left child here, then delete that position (in remove_value()
        // below). We can get to the largest value from our left child by
        // decrementing iter.
        iterator internal_iter(iter);
        --iter;
        assert(iter.node->leaf());
        params_type::move(mutable_allocator(), iter.node->slot(iter.position),
                          internal_iter.node->slot(internal_iter.position));
        internal_delete = true;
    }

    // Delete the key from the leaf.
    iter.node->remove_value(iter.position, mutable_allocator());
    --size_;

    // We want to return the next value after the one we just erased. If we
    // erased from an internal node (internal_delete == true), then the next
    // value is ++(++iter). If we erased from a leaf node (internal_delete ==
    // false) then the next value is ++iter. Note that ++iter may point to an
    // internal node and the value in the internal node may move to a leaf node
    // (iter.node) when rebalancing is performed at the leaf level.

    iterator res = rebalance_after_delete(iter);

    // If we erased from an internal node, advance the iterator.
    if (internal_delete) {
        ++res;
    }
    return res;
}

template<typename P>
auto btree<P>::rebalance_after_delete(iterator iter) -> iterator {
    // Merge/rebalance as we walk back up the tree.
    iterator res(iter);
    bool first_iteration = true;
    for (;;) {
        if (iter.node == root()) {
            try_shrink();
            if (empty()) {
                return end();
            }
            break;
        }
        if (iter.node->count() >= kMinNodeValues) {
            break;
        }
        bool merged = try_merge_or_rebalance(&iter);
        // On the first iteration, we should update `res` with `iter` because `res`
        // may have been invalidated.
        if (first_iteration) {
            res = iter;
            first_iteration = false;
        }
        if (!merged) {
            break;
        }
        iter.position = iter.node->position();
        iter.node = iter.node->parent();
    }

    // Adjust our return value. If we're pointing at the end of a node, advance
    // the iterator.
    if (res.position == res.node->count()) {
        res.position = res.node->count() - 1;
        ++res;
    }

    return res;
}

template<typename P>
auto btree<P>::erase(iterator begin, iterator end)
-> std::pair<size_type, iterator> {
    difference_type count = std::distance(begin, end);
    assert(count >= 0);

    if (count == 0) {
        return {0, begin};
    }

    if (count == size_) {
        clear();
        return {count, this->end()};
    }

    if (begin.node == end.node) {
        erase_same_node(begin, end);
        size_ -= count;
        return {count, rebalance_after_delete(begin)};
    }

    const size_type target_size = size_ - count;
    while (size_ > target_size) {
        if (begin.node->leaf()) {
            const size_type remaining_to_erase = size_ - target_size;
            const size_type remaining_in_node = begin.node->count() - begin.position;
            begin = erase_from_leaf_node(
                    begin, (std::min)(remaining_to_erase, remaining_in_node));
        } else {
            begin = erase(begin);
        }
    }
    return {count, begin};
}

template<typename P>
void btree<P>::erase_same_node(iterator begin, iterator end) {
    assert(begin.node == end.node);
    assert(end.position > begin.position);

    node_type *node = begin.node;
    size_type to_erase = end.position - begin.position;
    if (!node->leaf()) {
        // Delete all children between begin and end.
        for (size_type i = 0; i < to_erase; ++i) {
            internal_clear(node->child(begin.position + i + 1));
        }
        // Rotate children after end into new positions.
        for (size_type i = begin.position + to_erase + 1; i <= node->count(); ++i) {
            node->set_child(i - to_erase, node->child(i));
            node->clear_child(i);
        }
    }
    node->remove_values_ignore_children(begin.position, to_erase,
                                        mutable_allocator());

    // Do not need to update rightmost_, because
    // * either end == this->end(), and therefore node == rightmost_, and still
    //   exists
    // * or end != this->end(), and therefore rightmost_ hasn't been erased, since
    //   it wasn't covered in [begin, end)
}

template<typename P>
auto btree<P>::erase_from_leaf_node(iterator begin, size_type to_erase)
-> iterator {
    node_type *node = begin.node;
    assert(node->leaf());
    assert(node->count() > begin.position);
    assert(begin.position + to_erase <= node->count());

    node->remove_values_ignore_children(begin.position, to_erase,
                                        mutable_allocator());

    size_ -= to_erase;

    return rebalance_after_delete(begin);
}

template<typename P>
template<typename K>
auto btree<P>::erase_unique(const K &key) -> size_type {
    const iterator iter = internal_find(key);
    if (iter.node == nullptr) {
        // The key doesn't exist in the tree, return nothing done.
        return 0;
    }
    erase(iter);
    return 1;
}

template<typename P>
template<typename K>
auto btree<P>::erase_multi(const K &key) -> size_type {
    const iterator begin = internal_lower_bound(key);
    if (begin.node == nullptr) {
        // The key doesn't exist in the tree, return nothing done.
        return 0;
    }
    // Delete all of the keys between begin and upper_bound(key).
    const iterator end = internal_end(internal_upper_bound(key));
    return erase(begin, end).first;
}

template<typename P>
void btree<P>::clear() {
    if (!empty()) {
        internal_clear(root());
    }
    mutable_root() = EmptyNode();
    rightmost_ = EmptyNode();
    size_ = 0;
}

template<typename P>
void btree<P>::swap(btree &x) {
    using std::swap;
    if (abel::allocator_traits<
            allocator_type>::propagate_on_container_swap::value) {
        // Note: `root_` also contains the allocator and the key comparator.
        swap(root_, x.root_);
    } else {
        // It's undefined behavior if the allocators are unequal here.
        assert(allocator() == x.allocator());
        swap(mutable_root(), x.mutable_root());
        swap(*mutable_key_comp(), *x.mutable_key_comp());
    }
    swap(rightmost_, x.rightmost_);
    swap(size_, x.size_);
}

template<typename P>
void btree<P>::verify() const {
    assert(root() != nullptr);
    assert(leftmost() != nullptr);
    assert(rightmost_ != nullptr);
    assert(empty() || size() == internal_verify(root(), nullptr, nullptr));
    assert(leftmost() == (++const_iterator(root(), -1)).node);
    assert(rightmost_ == (--const_iterator(root(), root()->count())).node);
    assert(leftmost()->leaf());
    assert(rightmost_->leaf());
}

template<typename P>
void btree<P>::rebalance_or_split(iterator *iter) {
    node_type *&node = iter->node;
    int &insert_position = iter->position;
    assert(node->count() == node->max_count());
    assert(kNodeValues == node->max_count());

    // First try to make room on the node by rebalancing.
    node_type *parent = node->parent();
    if (node != root()) {
        if (node->position() > 0) {
            // Try rebalancing with our left sibling.
            node_type *left = parent->child(node->position() - 1);
            assert(left->max_count() == kNodeValues);
            if (left->count() < kNodeValues) {
                // We bias rebalancing based on the position being inserted. If we're
                // inserting at the end of the right node then we bias rebalancing to
                // fill up the left node.
                int to_move = (kNodeValues - left->count()) /
                              (1 + (insert_position < kNodeValues));
                to_move = (std::max)(1, to_move);

                if (((insert_position - to_move) >= 0) ||
                    ((left->count() + to_move) < kNodeValues)) {
                    left->rebalance_right_to_left(to_move, node, mutable_allocator());

                    assert(node->max_count() - node->count() == to_move);
                    insert_position = insert_position - to_move;
                    if (insert_position < 0) {
                        insert_position = insert_position + left->count() + 1;
                        node = left;
                    }

                    assert(node->count() < node->max_count());
                    return;
                }
            }
        }

        if (node->position() < parent->count()) {
            // Try rebalancing with our right sibling.
            node_type *right = parent->child(node->position() + 1);
            assert(right->max_count() == kNodeValues);
            if (right->count() < kNodeValues) {
                // We bias rebalancing based on the position being inserted. If we're
                // inserting at the beginning of the left node then we bias rebalancing
                // to fill up the right node.
                int to_move =
                        (kNodeValues - right->count()) / (1 + (insert_position > 0));
                to_move = (std::max)(1, to_move);

                if ((insert_position <= (node->count() - to_move)) ||
                    ((right->count() + to_move) < kNodeValues)) {
                    node->rebalance_left_to_right(to_move, right, mutable_allocator());

                    if (insert_position > node->count()) {
                        insert_position = insert_position - node->count() - 1;
                        node = right;
                    }

                    assert(node->count() < node->max_count());
                    return;
                }
            }
        }

        // Rebalancing failed, make sure there is room on the parent node for a new
        // value.
        assert(parent->max_count() == kNodeValues);
        if (parent->count() == kNodeValues) {
            iterator parent_iter(node->parent(), node->position());
            rebalance_or_split(&parent_iter);
        }
    } else {
        // Rebalancing not possible because this is the root node.
        // Create a new root node and set the current root node as the child of the
        // new root.
        parent = new_internal_node(parent);
        parent->init_child(0, root());
        mutable_root() = parent;
        // If the former root was a leaf node, then it's now the rightmost node.
        assert(!parent->child(0)->leaf() || parent->child(0) == rightmost_);
    }

    // Split the node.
    node_type *split_node;
    if (node->leaf()) {
        split_node = new_leaf_node(parent);
        node->split(insert_position, split_node, mutable_allocator());
        if (rightmost_ == node) rightmost_ = split_node;
    } else {
        split_node = new_internal_node(parent);
        node->split(insert_position, split_node, mutable_allocator());
    }

    if (insert_position > node->count()) {
        insert_position = insert_position - node->count() - 1;
        node = split_node;
    }
}

template<typename P>
void btree<P>::merge_nodes(node_type *left, node_type *right) {
    left->merge(right, mutable_allocator());
    if (right->leaf()) {
        if (rightmost_ == right) rightmost_ = left;
        delete_leaf_node(right);
    } else {
        delete_internal_node(right);
    }
}

template<typename P>
bool btree<P>::try_merge_or_rebalance(iterator *iter) {
    node_type *parent = iter->node->parent();
    if (iter->node->position() > 0) {
        // Try merging with our left sibling.
        node_type *left = parent->child(iter->node->position() - 1);
        assert(left->max_count() == kNodeValues);
        if ((1 + left->count() + iter->node->count()) <= kNodeValues) {
            iter->position += 1 + left->count();
            merge_nodes(left, iter->node);
            iter->node = left;
            return true;
        }
    }
    if (iter->node->position() < parent->count()) {
        // Try merging with our right sibling.
        node_type *right = parent->child(iter->node->position() + 1);
        assert(right->max_count() == kNodeValues);
        if ((1 + iter->node->count() + right->count()) <= kNodeValues) {
            merge_nodes(iter->node, right);
            return true;
        }
        // Try rebalancing with our right sibling. We don't perform rebalancing if
        // we deleted the first element from iter->node and the node is not
        // empty. This is a small optimization for the common pattern of deleting
        // from the front of the tree.
        if ((right->count() > kMinNodeValues) &&
            ((iter->node->count() == 0) ||
             (iter->position > 0))) {
            int to_move = (right->count() - iter->node->count()) / 2;
            to_move = (std::min)(to_move, right->count() - 1);
            iter->node->rebalance_right_to_left(to_move, right, mutable_allocator());
            return false;
        }
    }
    if (iter->node->position() > 0) {
        // Try rebalancing with our left sibling. We don't perform rebalancing if
        // we deleted the last element from iter->node and the node is not
        // empty. This is a small optimization for the common pattern of deleting
        // from the back of the tree.
        node_type *left = parent->child(iter->node->position() - 1);
        if ((left->count() > kMinNodeValues) &&
            ((iter->node->count() == 0) ||
             (iter->position < iter->node->count()))) {
            int to_move = (left->count() - iter->node->count()) / 2;
            to_move = (std::min)(to_move, left->count() - 1);
            left->rebalance_left_to_right(to_move, iter->node, mutable_allocator());
            iter->position += to_move;
            return false;
        }
    }
    return false;
}

template<typename P>
void btree<P>::try_shrink() {
    if (root()->count() > 0) {
        return;
    }
    // Deleted the last item on the root node, shrink the height of the tree.
    if (root()->leaf()) {
        assert(size() == 0);
        delete_leaf_node(root());
        mutable_root() = EmptyNode();
        rightmost_ = EmptyNode();
    } else {
        node_type *child = root()->child(0);
        child->make_root();
        delete_internal_node(root());
        mutable_root() = child;
    }
}

template<typename P>
template<typename IterType>
ABEL_FORCE_INLINE IterType btree<P>::internal_last(IterType iter) {
    assert(iter.node != nullptr);
    while (iter.position == iter.node->count()) {
        iter.position = iter.node->position();
        iter.node = iter.node->parent();
        if (iter.node->leaf()) {
            iter.node = nullptr;
            break;
        }
    }
    return iter;
}

template<typename P>
template<typename... Args>
ABEL_FORCE_INLINE auto btree<P>::internal_emplace(iterator iter, Args &&... args)
-> iterator {
    if (!iter.node->leaf()) {
        // We can't insert on an internal node. Instead, we'll insert after the
        // previous value which is guaranteed to be on a leaf node.
        --iter;
        ++iter.position;
    }
    const int max_count = iter.node->max_count();
    if (iter.node->count() == max_count) {
        // Make room in the leaf for the new item.
        if (max_count < kNodeValues) {
            // Insertion into the root where the root is smaller than the full node
            // size. Simply grow the size of the root node.
            assert(iter.node == root());
            iter.node =
                    new_leaf_root_node((std::min<int>)(kNodeValues, 2 * max_count));
            iter.node->swap(root(), mutable_allocator());
            delete_leaf_node(root());
            mutable_root() = iter.node;
            rightmost_ = iter.node;
        } else {
            rebalance_or_split(&iter);
        }
    }
    iter.node->emplace_value(iter.position, mutable_allocator(),
                             std::forward<Args>(args)...);
    ++size_;
    return iter;
}

template<typename P>
template<typename K>
ABEL_FORCE_INLINE auto btree<P>::internal_locate(const K &key) const
-> SearchResult<iterator, is_key_compare_to::value> {
    return internal_locate_impl(key, is_key_compare_to());
}

template<typename P>
template<typename K>
ABEL_FORCE_INLINE auto btree<P>::internal_locate_impl(
        const K &key, std::false_type /* IsCompareTo */) const
-> SearchResult<iterator, false> {
    iterator iter(const_cast<node_type *>(root()), 0);
    for (;;) {
        iter.position = iter.node->lower_bound(key, key_comp()).value;
        // NOTE: we don't need to walk all the way down the tree if the keys are
        // equal, but determining equality would require doing an extra comparison
        // on each node on the way down, and we will need to go all the way to the
        // leaf node in the expected case.
        if (iter.node->leaf()) {
            break;
        }
        iter.node = iter.node->child(iter.position);
    }
    return {iter};
}

template<typename P>
template<typename K>
ABEL_FORCE_INLINE auto btree<P>::internal_locate_impl(
        const K &key, std::true_type /* IsCompareTo */) const
-> SearchResult<iterator, true> {
    iterator iter(const_cast<node_type *>(root()), 0);
    for (;;) {
        SearchResult<int, true> res = iter.node->lower_bound(key, key_comp());
        iter.position = res.value;
        if (res.match == MatchKind::kEq) {
            return {iter, MatchKind::kEq};
        }
        if (iter.node->leaf()) {
            break;
        }
        iter.node = iter.node->child(iter.position);
    }
    return {iter, MatchKind::kNe};
}

template<typename P>
template<typename K>
auto btree<P>::internal_lower_bound(const K &key) const -> iterator {
    iterator iter(const_cast<node_type *>(root()), 0);
    for (;;) {
        iter.position = iter.node->lower_bound(key, key_comp()).value;
        if (iter.node->leaf()) {
            break;
        }
        iter.node = iter.node->child(iter.position);
    }
    return internal_last(iter);
}

template<typename P>
template<typename K>
auto btree<P>::internal_upper_bound(const K &key) const -> iterator {
    iterator iter(const_cast<node_type *>(root()), 0);
    for (;;) {
        iter.position = iter.node->upper_bound(key, key_comp());
        if (iter.node->leaf()) {
            break;
        }
        iter.node = iter.node->child(iter.position);
    }
    return internal_last(iter);
}

template<typename P>
template<typename K>
auto btree<P>::internal_find(const K &key) const -> iterator {
    auto res = internal_locate(key);
    if (res.HasMatch()) {
        if (res.IsEq()) {
            return res.value;
        }
    } else {
        const iterator iter = internal_last(res.value);
        if (iter.node != nullptr && !compare_keys(key, iter.key())) {
            return iter;
        }
    }
    return {nullptr, 0};
}

template<typename P>
void btree<P>::internal_clear(node_type *node) {
    if (!node->leaf()) {
        for (int i = 0; i <= node->count(); ++i) {
            internal_clear(node->child(i));
        }
        delete_internal_node(node);
    } else {
        delete_leaf_node(node);
    }
}

template<typename P>
int btree<P>::internal_verify(
        const node_type *node, const key_type *lo, const key_type *hi) const {
    assert(node->count() > 0);
    assert(node->count() <= node->max_count());
    if (lo) {
        assert(!compare_keys(node->key(0), *lo));
    }
    if (hi) {
        assert(!compare_keys(*hi, node->key(node->count() - 1)));
    }
    for (int i = 1; i < node->count(); ++i) {
        assert(!compare_keys(node->key(i), node->key(i - 1)));
    }
    int count = node->count();
    if (!node->leaf()) {
        for (int i = 0; i <= node->count(); ++i) {
            assert(node->child(i) != nullptr);
            assert(node->child(i)->parent() == node);
            assert(node->child(i)->position() == i);
            count += internal_verify(
                    node->child(i),
                    (i == 0) ? lo : &node->key(i - 1),
                    (i == node->count()) ? hi : &node->key(i));
        }
    }
    return count;
}

}  // namespace container_internal

}  // namespace abel

#endif  // ABEL_CONTAINER_INTERNAL_BTREE_H_
